tenthclass content em mathematics 04 linearprogramming

8/11/2019 .. TenthClass Content EM Mathematics 04 LinearProgramming

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This chapter comes under Paper -I. From this chapter 5 mark questions -1 (15=5m), 4marks questions - 1 (14=4m), 2 Mark Questions - 1 (1 2=2M) and 1 Mark Question - 1 (11=1M) and 6 objective bits (61/2=3M) altogether we can score 15 Marks easily. Heretheory part muset be read by students fully to attempt 1 mark question and objective bits.The information given below will help the students who are going to appear public exams.

Convex set: If P, Q X then the plane X is called convex set (or) If the line segmentoining any two points of a plane entirely lies in the same plane, then the plane is calledconvex set.

e.g.

are convex sets.

are not the convex sets.

PQ

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Linear Programming

etc

etc


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1. Indicate the polygonal region represented by the system of inequations x 1, y

1, x 3, y 3. 2M

Sol:i) x 1 boundary line x = 1 which is parallel to y-axis and shaded region is towards rightside of the lineii) y 1 boundary line is y=1 which is parallel to x-axis and shaded region is upwards forthe lineiii) x 3 boundary line is x=3 which is parallel to x-axis and shaded region is towards leftside of the line.iv) y 3 boundary line is y=3 which is parallel to x-axis and shaded region is downwardsfor the line

Here the four shaded region is the solution set for the given system of inequations.

2. Indicate the polygonal region represented by the system of ineqauation x 0, y 0, x + y 1 2M

Sol:i) x 0 boundary line is x=0 which represents y-axis and shaded region should be rightside of y-axis

ii) y 0 boundary line is y=0 which represents x-axis and shaded region should beupwards for x-axis

iii) x + y 1 boundary line is x+y = 1x + y = 1put x = 0 y = 1 (0, 1)put y = 0 x = 1 (1, 0)(0, 1), (1, 0)




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(0, 0): x + y 10 + 0 10 1 (True) Shaded region should be towards the origin including boundary line.

Here triple shaded region is the solution set

Linear Programming Problem (L.P.P.):- Which contains an objective function subject to certain constraints which are expressed

in the form of Linear inequation is called Linear Programming Problem.

Objective Function :- A function which is to be maximised or minimised in L.P.P is called an objective function

or Profit function.

Note: The solution of Linear inequation is either closed convex polygon or open convex

polygon.

The Fundamental Theorem :

- The objective function of L.P.P will be maximised or minimised at any are of the verticesof Convex polygon.

Feasible region :- The region formed by the constrains of L.P.P is called Feasible region.

Feasible Solution (L.P.P.):- Every point in Feasible region of L.P.P. is called Feasible Solution.

Optimum Solution :- The Solution which makes the objective function either minimum or maximum is called




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"Optimum Solution".

Iso-Profit Lines:- The system of Parallel lines obtained form the objective function in L.P.P. are called

Isoprofit lines.

Properties of isoprofit lines:1. Iso-Profit lines are parallel to each other.2. If iso-profit line co-incides with the boundary of convex polygon then the LPP has infinite

soluiton.3. No point in the feasible region makes the objective function either maximum or minimum

then the no.of solutin of Lpp are 04. If the isoprofit line moves away from the origin then the profit will be increased.5. If the iso-profit line moves towards the origin then the profit will be minimised.

3. A sweet shop makes gift packets of sweet combines two special types of sweets

A and B which weigh 7kg. Atleast 3kg of A and no more than 5 kgs of B should beused. The shop makes a profit of Rs. 15 on A and Rs 20 on B per kg. Determine theproduct mix so as to obtain maximum profit. [Construct the Linear ProgrammingProblem, graph is not required]. 4M

Sol:These are two types of sweets i.e. A and B.let the weight 'A' type sweets be x kgs and 'B' type sweets be y kgs. x 0, y 0 ...............(1)The Packet weight should not increase 7 kgs.

x + y

7 ..............(2)In each packet atleast 3 kgs of A type and not more than 5 kgs of B type should be usedi.e. x 3 .................(3)and y 5 ..................(4)Profit on 'A' type sweets per kg is Rs 15 and 'B' type sweets per kg is Rs. 20 Objective function or profit functionf = 15x + 20y

The linear programming problem is

Maximise f = 15x + 20y subject to the Constrants x+y 7, x 3, y 5, x 0, y 0.

4. Maximinse f =2x + y subject to the constraints 2x+y 8, y 4, x 3, x 0 and y 0

Sol:

i) x 0, y 0 represents 1st quadrantii) y 4, boundary line is y = 4iii) x 3 boundary line is x = 3iv) 2x+y 8, boundary line is 2x + y = 8

put x= 0 2(0) + y = 8 0 + y = 8 y = 8 (0, 8)put y = 0 2x + 0 = 8 2x = 8 x = 8/2 x = 4 (4, 0){(0, 8); (4, 0)}

(0, 0): 2(0) + 0 8 0 + 0 8 0 8 (F)




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shaded region is on the other side of origin.

Vertices of polygon are (3, 2); (3, 4); (2, 4)f = 2x + y;at (3, 2); f = 2(3) + 2 = 6 + 2 = 8at (3, 4); f = 2(3) + 4 = 6 + 4 = 10at (2, 4); f = 2(2) + 4 = 4 + 4 = 8

Maximum value of f is 10 at (3, 4)

5. Minimise f = x+y, subject to the constraints x+y 6; 2x+y 8, x 0, and y 0.sol:

f = x+y; x + y 6; 2x+y 8; x 0 and y 0

i) x 0; y 0 represents 1st Quadrantii) x + y 6; boundary line is x+y = 6put x = 0 0 + y = 6 y = 6 (0, 6)put y = 0 x + 0 = 6 x = 6 (6, 0)= {(0, 6); (6, 0)}




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(0, 0): 0 + 0 6 0 6 (F) Shaded region is one the other side of the origin

ii) 2x + y 8, boundary line is 2x + y = 8put x = 0 2(0) + y = 8 0 + y = 8 y = 8 (0, 8)put y = 0 2x + 0 = 8 2x = 8 x = 8/2 x = 4 (4, 0){(0, 8); (4, 0)}(0, 0): 2(0) + 0 8 0 + 0 8 0 8 (F) Shaded region is on the other side of the origin.Vertices of polygon are (0, 8), (2, 4), (6,0)at (0, 8); f = x+y f = 0 + 8 f = 8at (2, 4); f = x+y f = 2 + 4 f = 6at (6, 0); f = x+y f = 6 + 0 f = 6Minimum Value of f is 6 at (2, 4); (6, 0)




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Objective bits

1. If x > 0, y < 0 then (x, y) lies in ______ quadrant.

In the adjacent figure the graph represents _______3. The solution set of x y and x y is _______4. The line x=k is _______ to y-axis and 'k' units away from _______ axis.5. The y=0 represents _______ axis6. The solution set of Linear programming problem lies in _______ quadrant.7. If y=mx+c passes through origin, C = _______

8. If a < 0 and a

R then (a, a) lies in _______ quadrant.9. The value of f = 2x + 3y at the intersecting point of x = 2 and y = 3 _______10.If f = 3x + 4y is an objective function, the equatin of isoprofit line which passes through

(3, 4) is _______ or _______

Answers:

1. IV2. x + y 33. x = y

4. Parallel, y - axis, y-axis5. x6. I

7. 08. IV9. 1310. 3x+4y = 25, 3x+4y25 = 0

Assignment:

1. Define ''iso-profit line" 1M2. Indicate the polygonal region represented by the system of inequation x0, y4 and x y

2M3. Maximise f = 5x + 7y subject to the constraints 2x + 3y 12, 3x + y 12, x 0 and y 0

5M

4. Minimise f=3x+2y, subject to the constraints x + y 1, x y, 0 x 1, y 0. 5M

5. A shop keeper sells not more than 30 shirts of each colour. At least twice as many whiteones are sold as green ones. If the profit on each of the white be Rs. 20 and that of greenbe Rs. 25, how many of each kind be sold to give him a maximum profit? [Construct only

the Linear Programming Problem] 4M



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tenthclass content em mathematics 04 linearprogramming

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